The torsion in tensor product of modules was initially studied by Auslander in his seminal paper “Modules over unramified regular local rings" [1]}. For nonzero finitely generated modules \(M\)  and \(N\)  over an unramified (or equi-characteristic) regular local ring, Auslander proved that \(M\)  and \(N\)  are torsion-free and \(\operatorname{Tor}_i^R(M,N)=0\)  for all \(i\ge 1\) , provided that \(M\otimes _RN\)  is torsion-free; subsequently, the ramified case of Auslander's result was established by Lichtenbaum [2]}. Theorem REF  provides a generalization of the aforementioned results of Auslander and Lichtenbaum; this is because the torsion submodule of a module \(M\)  can be characterized by the \(\operatorname{\mathcal {Z}}\) -torsion submodule \(\Gamma _{\operatorname{\mathcal {Z}}}(M)\)  of \(M\)  for a suitable choice of \(\operatorname{\mathcal {Z}}\) ; see Proposition REF  and Corollary REF . To the best of our knowledge, the conclusion of Theorem REF  is new even for the closed subsets of \(\operatorname{Spec}(R)\) .
